If you take a simplicial set and ‘throw away’ the last face and degeneracy, and relabel, shifting everything down one ‘notch’, you get a new simplicial set. This is what is called the décalage of a simplicial set.
It is a model for the path space object of , –or rather: the union of all based path space objects for all basepoints – similar to, but a little smaller than, the model , which is discussed for instance at factorization lemma:
In the latter case an -cell in the path space is a morphism to from the simplicial cone over the -simplex modeled as the pushout . This is the simplicial set obtained by forming the simplicial cylinder over and then contracting one end to the point.
Contrary to that, an -simplex in the décalage of is a morphism to from the cone over modeled simply by the join of simplicial sets .
This is a much smaller model for the cone. In fact is just the -simplex. On the other hand, the above pushout-construction produces simplicial sets with many -simplices, the one that one “expects”, but glued to others with some degenerate edges. Accordingly, there is, for , a proper inclusion
As a result, the décalage construction is often more convenient than forming .
A central application is the special case where is the simplicial delooping of a simplicial group (see at simplicial principal bundle). In this case , called , is a standard model for the universal simplicial principal bundle.
The plain definition of the décalage of a simplicial set is very simple, stated below in
However, in order to appreciate and handle this definition, it is useful to understand it as a special case of total décalage, stated below in
From this one sees more manifestly that the décalage of a simplicial set is built from cones in the original simplicial set. This we discuss below in
In this last formulation it is clearest what the two canonical morphisms out of the décalage of a simplicial set mean. These we define in
Concretely, the décalage construction is the following.
For a simplicial set, the décalage of , is the simplicial set obtained by shifting every dimension down by one, ‘forgetting’ the last face and degeneracy of in each dimension:
It is often useful to understand this as a special case of the total décalage construction:
for the box product functor that takes to the bisimplicial set
This appears as (Stevenson 12, lemma 2.1).
It follows that the left adjoint of plain décalage forms joins with the 0-simplex:
The left adjoint to is
In particular for connected we have
This appears as (Stevenson 12, cor. 2.1).
By adjunction we have for all
So this exhibits the -cells of as being the cones of -simplices in .
For its décalage comes with two canonical morphisms out of it
Here in terms of the description above of décalage by cones:
the horizontal morphism is induced from the canonical inclusion ;
the vertical morphism is given by the canonical inclusion .
Or in terms of components, as discussed above,
the horizontal morphism is given by , hence in degree by the remaining face map ;
the vertical morphism is given in degree 0 by and in every higher degree similarly by .
See for instance (Stevenson, around def. 2) for an account.
For a simplicial set, the two morphisms from prop. 1 have the following properties.
If is a Kan complex, then
The first statement is classical, it appears for instance as (Stevenson 11, lemma 5).
For the second, notice that by remark 2 the lifting problem
is equivalent to the lifting problem
For each of these the statement that the projection is a Kan fibration if is a Kan complex, and moreover that it is a a right fibration if is a quasi-category, is (Joyal, theorem 3.19), reproduced also as (HTT, prop. 18.104.22.168). Notice that left/right fibrations into a Kan complex are automatically Kan fibrations (by the discussion at Left fibration in ∞-groupoids).
there is a diagram
The inclusion presents a canonical effective epimorphism in an (∞,1)-category in ∞Grpd into , out of a 0-truncated object. By the above, the décalage is a natural fibration resolution of this canonical “atlas”.
This is useful for instance in the discussion of homotopy pullbacks of this effective epimorphism: by the discussion there the homotopy pullback of along any morphism is presented by the ordinary pullback of any Kan fibration resolution, hence in particular of the décalage projection:
Décalage also has an abstract category theoretic description as follows. The simplex category, as a monoidal category equipped with the monoid , is the “walking monoid”, i.e., is initial among monoidal categories equipped with a monoid. Therefore is the walking comonoid; as a result, there is a comonad
which induces a comonad on simplicial sets whose underlying functor is precisely décalage:
The map is the counit of this comonad. The comonad itself is analogous to a kind of unbased path space object comonad on whose value at a space is a pullback
where is the set-theoretic identity inclusion of equipped with the discrete topology. Thus we have
the sum over all possible basepoints of path spaces based at . The analogy is made precise by a canonical isomorphism
where is simplicial singularization.
A -coalgebra partitions into path components and exhibits contractibility of each component. Similarly, a coalgebra of the decelage comonad exhibits the acyclicity of the underlying simplicial set.
Original sources are
Reviews are in
The link with simplicial groups and algebraic models of homotopy -types is given in
Tim Porter, n-types of simplicial groups and crossed n-cubes, Topology, 32, (1993), 5–24.
A detailed account of various technical aspects is in
and in secton 2.2 of
Closely related technical results are in section 3 of
An application in the theory of stacks is discussed in